(untitled) (Analysis in Theory and Applications, 2012)
We consider the problem K(x)u_xx = u_tt, 0 < x < 1, t ≥ 0, with the boundary condition u(0, t) = g(t) L^2(R) and u_x(0, t) = 0, where K(x) is continuous and 0 < α ≤ K(x) < +∞. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on g. Considering the existence of a solution u(x, ·) H^2(R) and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore, we prove the uniqueness of the solution for this problem.